EEVblog 1479 - Is Your Calculator WRONG?

[EEVblog]

Is your calculator WRONG? It could be!
Looking at the issue of implied multiplication and how it can affect your calculations. Why do some Casio and TI calculators give a different result to others? And why do they differ from your phone calculator, google calculator, or Wolfram Alpha?
How Order of Operations matters.

[TheSteve]

As someone in the comments mentioned try Hiper Scientific Calculator or Hiper Calc Pro. Then you can pick which you'd prefer

It will even give a pop-up the first time you enter the expression asking which you want.

[RoGeorge]

The expressions to evaluate shown in the video thumbnail are not the same (pocket calculator screen vs. the Wolfram webpage), notice the extra parentheses in the calculator's LCD.

[wraper]

The expressions to evaluate shown in the video thumbnail are not the same (pocket calculator screen vs. the Wolfram webpage), notice the extra parentheses in the calculator's LCD.

Thanks Captain Obvious. However if you watched the video for at least a few minutes, you would know why.

[RoGeorge]

The expressions to evaluate shown in the video thumbnail are not the same (pocket calculator screen vs. the Wolfram webpage), notice the extra parentheses in the calculator's LCD.

Thanks Captain Obvious. However if you watched the video for at least a few minutes, you would know why.

Thanks, Captain Smart Ass.  That was feedback for Dave.  The big red "Oops" button suggests there's something wrong with that result, though the thumbnail does not capture any unexpected results, so the thumbnail won't incite one to watch the video.

[SeanB]

Yes, and a good reason to actually teach how to do basic mathematics without using a calculator, so the pupils actually learn that the calculations work on certain assumptions, and that merely using a calculator is not always going to be definitely right. You really need to know how the operations take place before you just use a calculator, so I often will do some form of paper based calculation for simple things, to keep the memory fresh. Easiest is to work out VAT in your head, as that can be reasonably simple to do, just using simple addition and shifting digits for the division side. That or simple resistors in parallel, or to calculate the value needed to drive a LED at some specific current.

[Cnoob]

The HP prime in Textbook entry mode gives the answer correctly as 1.
In Algebraic entry mode gives the answer 9.

[EEVblog]

The expressions to evaluate shown in the video thumbnail are not the same (pocket calculator screen vs. the Wolfram webpage), notice the extra parentheses in the calculator's LCD.

Yes I know, it wouldn't let me get that shot without the calculator changing it.

[EEVblog]

Thanks, Captain Smart Ass.  That was feedback for Dave.  The big red "Oops" button suggests there's something wrong with that result, though the thumbnail does not capture any unexpected results, so the thumbnail won't incite one to watch the video.

You watched it.
I just Irfanviewed the thumbnail, better now?

[RoGeorge]

Perfect now, IMHO. 

[ledtester]

Now that I'm aware of this difference I think I like the Classwiz way of handling implied multiplication. For instance, it makes it more convenient to enter a complicated denominator.

fwiw, SpeedCrunch also evaluates "6/2(2+1)" as 1.

[armandine2]

fwiw - The TI-nspire CX II-T CAS looks ok

[SiliconWizard]

When I saw the title, I thought, "cool, maybe some interesting rounding issues", then I saw the first image of the video and immediately thought "oh no, not that again!".
And sure enough, it was "that" again. *Sighs*

[TimFox]

I am a die-hard RPN user.  (I first learned symbolic logic using "forward" Łukasiewicz notation.)
When proselytizing engineers, I watched them use their algebraic calculators and realized that they found it too difficult to keep the () organized, and either used storage (memory) locations or even Post-It notes for intermediate results.

[IanB]

Order of operations is over-simplified for school children. It gets more subtle in the real world.

For instance, in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

If there is any possibility of misinterpretation, then parentheses or formatting should be used to make the intention clear. Nobody should be playing games with readers in professional documents.

[TimFox]

Order of operations is over-simplified for school children. It gets more subtle in the real world.

For instance, in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

If there is any possibility of misinterpretation, then parentheses or formatting should be used to make the intention clear. Nobody should be playing games with readers in professional documents.

When in doubt, add parantheses and brackets as required to avoid mis-interpretation.  The conventional order is { [ (  ) ] } .

[ve7vie]

I have a fx-991ES PLUS C, and it gives 1 as the answer for 6/2(2+1). The calculator was bought in Canada.  So does my CG10, which shows the parens it adds. Both machines give 9 for 6/2x(2+1). But my CG500 gives a syntax error without the x. No implication allowed!

[ve7vie]

That's why I always programmed in APL. The only rule is right-to-left precedence.

[Nominal Animal]

Is it an error that in general, the order of precedence for implicit multiplication is not well defined?  I don't think so.

Order of precedence in general is one of those things like the "I before E except after C" 'rule' in English.  Albeit it sounds good, it does not seize the crux of the matter.  In other words, the only problem here is that people are told that a rule exists, when it really is not a reliable rule at all.

Parentheses work, and should be used to indicate the order of precedence, unless an order has already been agreed upon.  The one taught at school is only a common one, not "the" only one used.  Calculators are a perfect example of this.

Personally, I would have emphasized more the fact that the conventions vary – not just US vs. the rest of the Western world.  That it is a convention and not really a rule at all.  Therefore, it's just one of those things that change from calculator/environment/programming language to the next, and can bite one in the ass if one assumes too much without checking the assumptions first.

[NiHaoMike]

(Some) Casio calculators are weird.

[Vtile]

If that calculator feature will ruin your day, don't look up the Excel quirks!

And use reverse polish notation.
6 2 2 1 + × ÷

Edit. Ps.
Or if you prefer nine
6 2 ÷ 2 1 + ×

...what about square root of negative number, how many insist that it doesn't have roots?

What about elementary math answer as two for square root of 4 .... where the negative plain did go to hide? Maybe it is only on paraboloid solving formula, hmmm.....

[bd139]

If that calculator feature will ruin your day, don't look up the Excel quirks!

And use reverse polish notation.
6 2 2 1 + × ÷

No truer things said on the internet.

I know a company that did their economic projection modelling on Excel and due to an actual bug in regression modelling lost a lot of money. Microsoft know it exists and won't fix it because it might break millions of spreadsheets worldwide.

And yes, RPN, because the input is never ambiguous. I'm actually writing an RPL-ish calculator...

Code: [Select]

$ echo "eng in in inv swap inv + inv out" > prog
$ echo "12e3 12e3" | ./pec prog
6e3

This is part of a streaming programming environment I am trying to build out as a commercial product.

[IanB]

It's not RPN, it's NPR: "Notation, Polish, Reverse" 

[Vtile]

It's not RPN, it's NPR: "Notation, Polish, Reverse" 

[Vtile]

Kind of amusing thing is that with rigorous interpretation of PODMAS you get 9 and with rigorous interpretation of BEMDAS you get 1. That is if you do not give juxtaposition any weight.

6÷(2+1)2

[golden_labels]

I know a company that did their economic projection modelling on Excel and due to an actual bug in regression modelling lost a lot of money. Microsoft know it exists and won't fix it because it might break millions of spreadsheets worldwide.

While I myself never seen that leading to any catastrophe, the widespread use of spreadsheets for statistical modelling in demographics and macroeconomy was always filling me with dread. Each time I received a model to work on, expressed as an Excel document, I was shivering. With author blissfully unaware of floating point calculation issues, rounding for display or that various “magical” textual entries may be interpreted not as they think they are. Chances of that introducing a substantial error are acceptably low and it wouldn’t undermine the entire model, but it is always smelly.

In their defense: using dedicated statistical/maths software would not automatically solve that. Tools like SAS or MATLAB are unauditable anyway. Open software is, but in the end those people lack knowledge to evaluate the implementation by themselves. But using a piece of software, that was never designed with that purpose in mind, with unwavering confidence is still scary.

[Neilm]

I just tried this on the app I use on my phone. The first thing it did was ask me which version did I want then saved that as a setting.

[NiHaoMike]

If that calculator feature will ruin your day, don't look up the Excel quirks!

Reverse engineering of a rather infamous Excel floating point bug:
https://web.archive.org/web/20181120193921/http://www.lomont.org/Math/Papers/2007/Excel2007/Excel2007Bug.pdf

[Scherms]

...in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

If there is any possibility of misinterpretation, then...

... HP-RPN to the rescue!



RPN 'Reverse Polish Notation' HP-41CX (above) evaluates in this order: start with parentheses first, then work outwards.
6/2(2+1)

2 ENTER 1 + 2 x 6 /
1.0000   



Now I use an HP-28S (below) in algebraic mode: 



'6/2(2+1)' >> Syntax Error     

Hence edit to...

'6/2x(2+1)' EVAL
9.0000 

or...

'6/(2x(2+1))' EVAL
1.0000   


Now in HP-28S RPN:
2 ENTER 1 + 2 x 6 /
1.0000   

 

Lesson: ...stick with 'explicit' RPN at all times to eliminate any 'implied' algebraic errors!

[IanB]

Lesson: ...stick with 'explicit' RPN at all times to eliminate any 'implied' algebraic errors!

Yes, except in RPN you determine the order of operations by your key entry.

"Say what to do, and do what I say."

For example:

6 ENTER 2 ÷ 2 ENTER 1 + ×
9.0000

So you can get any answer you wish to get 

[kjpye]

RPN 'Reverse Polish Notation' HP-41CX (above) evaluates in this order: start with parentheses first, then work outwards.
6/2(2+1)

2 ENTER 1 + 2 x 6 /
1.0000   

Which is only coincidentally the correct answer! That is the answer to calculating 2(2+1)/6.

Try 2 ENTER 1 + 2 × 6 X<->Y /

[IanB]

Which is only coincidentally the correct answer! That is the answer to calculating 2(2+1)/6.

Try 2 ENTER 1 + 2 × 6 X<->Y /

Ah, but this is why there is a stack. And why not everyone can deal with RPN:

6 ENTER 2 ENTER ENTER 1 + × ÷

[bd139]

Lesson: ...stick with 'explicit' RPN at all times to eliminate any 'implied' algebraic errors!

Yes, except in RPN you determine the order of operations by your key entry.

"Say what to do, and do what I say."

For example:

6 ENTER 2 ÷ 2 ENTER 1 + ×
9.0000

So you can get any answer you wish to get 

Only if you’re an idiot 

[Scherms]

Lesson: ...stick with 'explicit' RPN at all times to eliminate any 'implied' algebraic errors!

Yes, except in RPN you determine the order of operations by your key entry.

"Say what to do, and do what I say."

For example:

6 ENTER 2 ÷ 2 ENTER 1 + ×
9.0000

So you can get any answer you wish to get 

 


Eh, NO! 

With RPN you start within the parentheses first and then work outwards. It is not strictly 'left to right' like in your example.

Go on, admit that you're a TI guy, we'll understand... 

[Scherms]

Try this online HP15C RPN calculator yourself to confirm method.



https://hp15c.com/web/hp15c.html

 

[Scherms]

Is your calculator WRONG? It could be!
Looking at the issue of implied multiplication and how it can affect your calculations. Why do some Casio and TI calculators give a different result to others? And why do they differ from your phone calculator, google calculator, or Wolfram Alpha?
How Order of Operations matters.

 

TexASS Instruments!
...no wonder NASA went with HP...

[Vtile]

Eh, NO! 

With RPN you start within the parentheses first and then work outwards. It is not strictly 'left to right' like in your example.

The actual problem in this expression and evaluation is not "parentheses" first so much, but a pattern recognition of the reader. How does he/she/they interpret what is written on the expression. After all mathematics are whole lots of pattern recognition, when it comes to many things, expression solving is no exception on this.

ie.
6:2(2+1)
6/2(2+1)
6/(2+1)2

Same with the -1² = -1 (as I have understood that it is now somewhat standard form in printed math), while I doubt there is none who actually uses math as tool puts this as result when using a pen and paper with same marking on paper, in other words would use parenthesis as (-1)²=1. In the other hand this is answer logical since what it actually contains is as far as I know actually (-1)*1² as in following expression: a-(b-c)=a+(-b)+c=a+(-1)(b-c)=a+[-1b+((-1)²)c] ... which to my eyes are again one of those pesky implied multiplications btw.

..but again -1²=-1 is illogical in a way the algebra is thought in ie. with quadratic formula ... aX²+bX+c is many times written as 1X²-6X+8, where b=-6 and transferred to formula ... what should be written is actually 1X²+[(-1)6]X+8 where b=(-1)6  .... with this former "style" -1² is actually +(-1)²

summa summarum.. use your head and do test calculation(s) if context do not reveal the intended formatting ...

...it seems it is a bed time, one should not take a "quick" look of eevblog in the middle of night.

[Scherms]

Eh, NO! 

With RPN you start within the parentheses first and then work outwards. It is not strictly 'left to right' like in your example.

The actual problem in this expression and evaluation is not "parentheses" first so much but a pattern recognition of the reader. How does he/she/they interpret what is written on the expression.

The main problem is the machine interpretation to a standard mathematical FORM with regard to 'implied' functions.

 


And as this topic is about dodgy SCI calculators not applying the proper FORM solutions to algebraic formula entries... So to the real point it is when choosing a SCI calculator the explicit warning 'caveat emptor' applies!



Also there is only one solution to this equation...

6/2(2+1) = 1

or

\$\frac{6}{2(2+1)}\$ = 1

... and is a good test for when buying a real SCI calculator!

QED

[IanB]

Same with the -1² = -1 (as I have understood that it is now somewhat standard form in printed math)

I think there was at some point an attempt to separate the negative sign from the minus operator in typography, so that you could write, for example, 1 − ^-1 = 2

But it seems this did not get much traction.

[Scherms]

-1² = -1

WRONG! 

 (-1)² = 1

[bd139]

Did some tests on my daughter's Casio FX-CG50 and I'm quite impressed.

So I entered the usual test case here verbatim. When you hit EXE it actually shows what its interpretation of the input is in the input line by adding parenthesis where it made assumptions for you to validate



Of course being natural input it allows much less ambiguous entry method so it should be represented as this really:



I rather like this. Also does engineering units with SI multiplier suffixes:

[Vtile]

Same with the -1² = -1 (as I have understood that it is now somewhat standard form in printed math)

I think there was at some point an attempt to separate the negative sign from the minus operator in typography, so that you could write, for example, 1 − ^-1 = 2

But it seems this did not get much traction.

Honestly the typographical "solution" is just stupid (I have a stronger adjective in my head, but as this is family friendly environment "stupid" shall be sufficient). None will ever want to judge something like that, based on subtle differences in vertical line length and variation of position. 

[bd139]

Typographical solution is fine. Been using it for over 40 years on paper

[Scherms]

Typographical solution is fine. Been using it for over 40 years on paper

I've been using my HP-41CX for 34+ years...
...before that a K&E Keuffel & Esser DECI-LON 68 1100 Slide Rule





 

[Vtile]

Typographical solution is fine. Been using it for over 40 years on paper

As long you can chose the typeset.... as in paper and pen.

[wizard69]

The HP prime in Textbook entry mode gives the answer correctly as 1.
In Algebraic entry mode gives the answer 9.

The question is which is correct.   Is there some sort of science based calculation, that is a calculation that can be verified against physical evidence that one is correct over the other.   The reason I ask (mind your this was 45 years ago, we where taught that expressions in parens get evaluated first and then it is left to right respecting operator precedence.   At least that is the way I remember it, though the DaveCalc explanation made some sense if the operator to the left was addition or subtraction.

It would make for an interesting discussion if we could find an example from science or geometry where one of these two orderings is completely wrong.   It just seems odd that the reset of the world is doing it wrong.   It is interesting that a calculator like Qalculate! offers three ways to evaluate the expression when I typed it in.   I just can't see any reasoning where implicit multiplication with a set of parens returning a value would take precedence over the rest of the evaluation.   In the end that is what I see, the parens are effectively a function that returns a value to be plugged into the equation. 

[IanB]

The question is which is correct.

There isn't a correct answer. It isn't a test, or a puzzle, it is rather an attempt at communication. If there can be disagreement about what is being communicated, then the communication has failed. In that case, all answers are wrong.

[Scherms]

The question is which is correct.

There isn't a correct answer...
...all answers are wrong.

I'm a chemical engineer --

NOPE!  Not in this case... make up your mind you hypocrite!

...in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

Yes, there is only one FORM solution to this equation...

6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$ = [6 / [2 x [2 + 1]]] = 1


... and is a good test for when buying an accurate SCI calculator!

[Scherms]

Just to be clear...



Even old HP calcs 40 years old won't allow implicit syntax to be entered!

 

[wizard69]

The question is which is correct.

There isn't a correct answer...
...all answers are wrong.

I'm a chemical engineer --

NOPE!  Not in this case... make up your mind you hypocrite!

...in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

Yes, there is only one FORM solution to this equation...

6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$ = [6 / [2 x [2 + 1]]] = 1


... and is a good test for when buying an accurate SCI calculator!

I'm still up in the air here, I was always told to evaluate left to right respecting precedence.   Thus that would read as "six half times 3" or 3 * 3.   This would be the result of solving calculations within the parens and then restarting left to right.  I just find it amazing that half the world sees this differently.

This leaves me wondering how much screwed up software is out there.

[TimFox]

I suggest that careful workers either use sufficient parantheses/brackets to make their expressions unambiguous, or revert to variations on Łukasiewicz (Polish) notation where parantheses are not needed and there are no ambiguous formations.

[Scherms]

I suggest that careful workers either use sufficient parantheses/brackets to make their expressions unambiguous, or revert to variations on Łukasiewicz (Polish) notation where parantheses are not needed and there are no ambiguous formations.

This leaves me wondering how much screwed up software is out there.

That's where HP RPN comes in! It eliminates the confusing 'implicit' schemata... 

6 / 2(2 + 1)

Enter numbers onto stack:
6 ENTER, 2 ENTER, 2 ENTER, 1 ENTER



Enter operator:
+, x, /

= 1.0000

 

[Vtile]

...or eg. instruction list:

L 2.0
L 1.0
+R
L 2.0
*R
L 6.0
TAK
/R
T #RESULT

There is RPN outside HP Calcs.

[Scherms]

The DM41X, another member of our flagship line-up of RPN calculators, is a faithful recreation of the software environment of the HP-41CX, one of the most versatile calculators ever created and the first alphanumerical calculator, in a modern hardware architecture. Easily share or back up your programs or your entire DM41X environment and use hundreds of existing, professionally developed software applications with this rugged calculator sporting a stainless steel case and a Gorilla Glass screen.

Like all SwissMicros calculators, the DM41X uses the efficient and time-tested RPN logic.

[gnuarm]

The expressions to evaluate shown in the video thumbnail are not the same (pocket calculator screen vs. the Wolfram webpage), notice the extra parentheses in the calculator's LCD.

Thanks Captain Obvious. However if you watched the video for at least a few minutes, you would know why.

Why the need to be rude?  The real question is "why" do they use the two methods?  I didn't see that explained in the ever annoying video.

[Scherms]

The real question is "why" do they use the two methods?  I didn't see that explained in the ever annoying video.

There is no two methods, one is structurally wrong and can't be used!

eg: WRONG




eg: RIGHT



 

[Nominal Animal]

The real question is "why" do they use the two methods?

Because humans and their languages.

(Why are lead, lead, and lead pronounced differently in English?)

It is like standards.  One is never sufficient, because humans.

Specifically, here the problem is that some humans think that it is acceptable to omit multiplication operator in certain situations, but do not think it is necessary to define the priority of (operator order for) said implicit multiplication.  Purely a human foible.

[gnuarm]

The real question is "why" do they use the two methods?

Because humans and their languages.

(Why are lead, lead, and lead pronounced differently in English?)

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

It is like standards.  One is never sufficient, because humans.

Specifically, here the problem is that some humans think that it is acceptable to omit multiplication operator in certain situations, but do not think it is necessary to define the priority of (operator order for) said implicit multiplication.  Purely a human foible.

I think the priority is established, the question is, why are the two multiply operators handled differently?

[Nominal Animal]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I think the priority is established, the question is, why are the two multiply operators handled differently?

For the same reason a pint is approximately 568 ml, or approximately 551 ml, or approximately 473 ml, depending on who you ask and in which context: because humans.

Would it be so odd to have two different multiplication operators, × and ·, that have a different priority?  After all, - and - do: one is unary negation, and the other is binary subtraction, both using the same character/glyph.

[gnuarm]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I don't know what LEAD /lid/ is.  The pronunciation of this is like the word "lid".  I can't find any reference that indicates this is a pronunciation of LEAD. 

I think the priority is established, the question is, why are the two multiply operators handled differently?

For the same reason a pint is approximately 568 ml, or approximately 551 ml, or approximately 473 ml, depending on who you ask and in which context: because humans.

That is not a reason.  This is math, with rules and structures.  Without that, math literally doesn't exist.

Would it be so odd to have two different multiplication operators, × and ·, that have a different priority?  After all, - and - do: one is unary negation, and the other is binary subtraction, both using the same character/glyph.

The character does not matter.  They clearly have different functions.  This is a false analogy. 

Having two operators with the same functionality is silly and pointless.  There must be a usage that makes a higher priority for the implied multiplication significant, even if it's only convenience.  But there has to be a use case where it makes a difference.

[Nominal Animal]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I don't know what LEAD /lid/ is.  The pronunciation of this is like the word "lid".  I can't find any reference that indicates this is a pronunciation of LEAD. 

Here, with audio.

I think the priority is established, the question is, why are the two multiply operators handled differently?

For the same reason a pint is approximately 568 ml, or approximately 551 ml, or approximately 473 ml, depending on who you ask and in which context: because humans.

That is not a reason.  This is math, with rules and structures.  Without that, math literally doesn't exist.

It is, because we have two layers of definitions here.

The surface layer is the words, acronyms, symbols, and glyphs we use.  These mutate, change, and evolve constantly.  These are the ones you are fully allowed to play with; the only requirement in science and math is that you define them.  In math, this is notation.  There are many valid notations, and if you do complex work, you often end up inventing your own if a suitable one does not yet exist.

The deep layer is the things and rules being described.  These are not to be messed with willy-nilly, because we've discovered them only through hard work and critical peer review.

Math is the deep layer.  What we are talking about is the notation, the surface layer, that is completely up to humans to define.

It is no different than defining "elektroni" = "electron" and "sähkälehitu" = "electron" (which does imply "elektroni" = "sähkälehitu").
Or what base (radix) we use when writing numbers.

The answer to the question "Why would implicit multiplication have a different priority than explicit multiplication?" is that the human surface layer, the notation layer, is always in flux, and not something you can blindly rely on.  (Well, currently, we can somewhat rely on the priorities of explicit operators, and use parentheses to ensure a specific order, as many have pointed out earlier in this thread.)

Personally, I only use implicit multiplication between numeric constants and variables.  This is because I find \$2 \pi x\$ easier to read than \$2 \times \pi \times x\$.  I do believe that human perception detail is at the core of why two different notations for multiplication exist.  I also believe that it is most likely human shortsightedness –– specifically, "out of sight, out of mind" –– that has caused implicit multiplication being omitted when humans have agreed upon the priorities of operators.  The priorities of the operators being just another notational detail: the math itself does not change, even if the notation does.

Having two operators with the same functionality is silly and pointless.  There must be a usage that makes a higher priority for the implied multiplication significant, even if it's only convenience.  But there has to be a use case where it makes a difference.

That assumes humans are rational beings.  Evidence suggests otherwise.

[IanB]

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

There are only two standard pronunciations. One is with short 'e' sound, as in 'led', the other is with a long 'ee' sound, as in 'leed'.

If you consider regional accents, these two pronunciations can have varying vowel sounds, but they still correspond to one or the other of the two. For example, in New Zealand, 'led' can sound like 'lid', but that is just the common pronunciation of 'e' in that part of the world.

Things get really complicated if you start considering regional accents. For example, the standard British pronunciation of 'a' in 'cat' is a different sound from the American 'a' in the same word. And then the American 'a' sound, while different from British, also varies itself in different parts of the USA.

[gnuarm]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I don't know what LEAD /lid/ is.  The pronunciation of this is like the word "lid".  I can't find any reference that indicates this is a pronunciation of LEAD. 

Here, with audio.

Zero help.  There were THREE pronunciations provided with THREE definitions.  The above link is very hard to follow, but I only see two. 

I think the priority is established, the question is, why are the two multiply operators handled differently?

For the same reason a pint is approximately 568 ml, or approximately 551 ml, or approximately 473 ml, depending on who you ask and in which context: because humans.

That is not a reason.  This is math, with rules and structures.  Without that, math literally doesn't exist.

It is, because we have two layers of definitions here.

The surface layer is the words, acronyms, symbols, and glyphs we use.  These mutate, change, and evolve constantly.  These are the ones you are fully allowed to play with; the only requirement in science and math is that you define them.  In math, this is notation.  There are many valid notations, and if you do complex work, you often end up inventing your own if a suitable one does not yet exist.

The deep layer is the things and rules being described.  These are not to be messed with willy-nilly, because we've discovered them only through hard work and critical peer review.

Math is the deep layer.  What we are talking about is the notation, the surface layer, that is completely up to humans to define.

It is no different than defining "elektroni" = "electron" and "sähkälehitu" = "electron" (which does imply "elektroni" = "sähkälehitu").
Or what base (radix) we use when writing numbers.

The answer to the question "Why would implicit multiplication have a different priority than explicit multiplication?" is that the human surface layer, the notation layer, is always in flux, and not something you can blindly rely on.  (Well, currently, we can somewhat rely on the priorities of explicit operators, and use parentheses to ensure a specific order, as many have pointed out earlier in this thread.)

Personally, I only use implicit multiplication between numeric constants and variables.  This is because I find \$2 \pi x\$ easier to read than \$2 \times \pi \times x\$.  I do believe that human perception detail is at the core of why two different notations for multiplication exist.  I also believe that it is most likely human shortsightedness –– specifically, "out of sight, out of mind" –– that has caused implicit multiplication being omitted when humans have agreed upon the priorities of operators.  The priorities of the operators being just another notational detail: the math itself does not change, even if the notation does.

Having two operators with the same functionality is silly and pointless.  There must be a usage that makes a higher priority for the implied multiplication significant, even if it's only convenience.  But there has to be a use case where it makes a difference.

That assumes humans are rational beings.  Evidence suggests otherwise.

If you say so. 

[Nominal Animal]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I don't know what LEAD /lid/ is.  The pronunciation of this is like the word "lid".  I can't find any reference that indicates this is a pronunciation of LEAD. 

Here, with audio.

Zero help.  There were THREE pronunciations provided with THREE definitions.  The above link is very hard to follow, but I only see two. 

You wrote you cannot find any reference to IPA /lid/.  (The same page contains the two other IPA pronunciations, if you look carefully.)  I posted a link to the etymology and to the audio file matching the IPA pronunciation.  If that's zero help, well, you're welcome.

That assumes humans are rational beings.  Evidence suggests otherwise.

If you say so.

Who cares what I say?  Go read a blog post describing some of that evidence with references instead.

[IanB]

You wrote you cannot find any reference to IPA /lid/.  (The same page contains the two other IPA pronunciations, if you look carefully.)  I posted a link to the etymology and to the audio file matching the IPA pronunciation.  If that's zero help, well, you're welcome.

You provided a link to this page: https://en.wiktionary.org/wiki/lead

It describes TWO English pronunciations. One that sounds like "led" and one that sounds like "leed".

[Scherms]

...in the real world of mathematics and engineering, adjacent terms bind more tightly than operators. Therefore a/bc is understood to be \$\frac{a}{bc}\$

Similarly, 6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$

Yes, there is only one FORM solution to this equation...

6/2(2+1) would be read as \$\frac{6}{2(2+1)}\$ = [6 / [2 x [2 + 1]]] = 1

[gnuarm]

I know how lead and lead are pronounced, but what is lead?  I've never heard of that word.

IPA /lɛd/, /liːd/, /lid/.  It turns out one of those is American pronunciation, the other two English.  Dunno about aussies, though; there could be more (/lɛ:d/?)

I don't know what LEAD /lid/ is.  The pronunciation of this is like the word "lid".  I can't find any reference that indicates this is a pronunciation of LEAD. 

Here, with audio.

Zero help.  There were THREE pronunciations provided with THREE definitions.  The above link is very hard to follow, but I only see two. 

You wrote you cannot find any reference to IPA /lid/.  (The same page contains the two other IPA pronunciations, if you look carefully.)  I posted a link to the etymology and to the audio file matching the IPA pronunciation.  If that's zero help, well, you're welcome.

Calm down.  Relax.  Take a deep breath and exhale. 

I saw a claim of three separate words that are spelled the same.  I got the IPA pronunciation notation mixed up.  Ok?  I'm looking for someone to tell me what the third word is.  I know what lead and lead are, now what is lead?

That assumes humans are rational beings.  Evidence suggests otherwise.

If you say so.

Who cares what I say?  Go read a blog post describing some of that evidence with references instead.

If you say so.

[gnuarm]

You wrote you cannot find any reference to IPA /lid/.  (The same page contains the two other IPA pronunciations, if you look carefully.)  I posted a link to the etymology and to the audio file matching the IPA pronunciation.  If that's zero help, well, you're welcome.

You provided a link to this page: https://en.wiktionary.org/wiki/lead

It describes TWO English pronunciations. One that sounds like "led" and one that sounds like "leed".

I was asking what the third word is.  It's not about pronunciations.  I'm asking what the word is. 

[IanB]

I was asking what the third word is.  It's not about pronunciations.  I'm asking what the word is.

There are more than three.

As a noun:
1. The metal
2. An inside track or a prospect, as in a sales lead
3. A position at the front, as to be in the lead
4. A cord or a cable, as in a power lead

As  a verb:
1. To be in front, as in to lead the field
2. To go somewhere, as in the path leads to the village
3. To draw along, as in to lead a bull by the nose

As an adjective:
1. Being made of lead, an in a lead weight
2. Being ahead of others, as in the lead position

I'm sure there may be others, but these are the first that come to mind.

[gnuarm]

I was asking what the third word is.  It's not about pronunciations.  I'm asking what the word is.

There are more than three.

As a noun:
1. The metal
2. An inside track or a prospect, as in a sales lead
3. A position at the front, as to be in the lead
4. A cord or a cable, as in a power lead

As  a verb:
1. To be in front, as in to lead the field
2. To go somewhere, as in the path leads to the village
3. To draw along, as in to lead a bull by the nose

As an adjective:
1. Being made of lead, an in a lead weight
2. Being ahead of others, as in the lead position

I'm sure there may be others, but these are the first that come to mind.

Your examples come down to two words:

1) Lead, the metal itself or relating to it's properties.  Pronounced like "led"

2) Lead, in some manner, referring to ahead of others or being pulled by something ahead of you, pronounced with a long e, "leed". 

While there are multiple definitions of these two words, they are still only two words.  Words often have multiple definitions, but are still only one word. 

Still, someone posted that there were three words, with three pronunciations, all spelled "lead". 

I suppose this has dragged out as much as is meaningful.  I'm happy to say you are right and there are an unlimited number of words spelled lead.

[Nominal Animal]

I was asking what the third word is.  It's not about pronunciations.  I'm asking what the word is.

If you say so.

[IanB]

Your examples come down to two words:

1) Lead, the metal itself or relating to it's properties.  Pronounced like "led"

2) Lead, in some manner, referring to ahead of others or being pulled by something ahead of you, pronounced with a long e, "leed". 

While there are multiple definitions of these two words, they are still only two words.  Words often have multiple definitions, but are still only one word.

You are absolutely right, and we are both agreed on this. One word is related to the metal, and generally sounds like "led"; the other word is related to the action, and generally sounds like "leed".

Still, someone posted that there were three words, with three pronunciations, all spelled "lead". 

That someone is mistaken, and there are not three different words, nor are there three different pronunciations.

[gnuarm]

I was asking what the third word is.  It's not about pronunciations.  I'm asking what the word is.

If you say so.

I dug back and it was YOU!  lol

[Nominal Animal]

That someone is mistaken, and there are not three different words, nor are there three different pronunciations.

If you say so.

It's not like I'm a native English speaker myself, you see.

I dug back and it was YOU!  lol

lol

[IanB]

That someone is mistaken, and there are not three different words, nor are there three different pronunciations.

If you say so.

I do say so.

[sleemanj]

I have no idea what all this talk of lead is,

[Nominal Animal]

I do say so.

Granted, and thanks for the reference.   The one I had at hand was wiktionary, and that specifies the third one as a misspelling of led (as past tense and past participle of lead).

I must say I find it funny how similar this "lead" discussion is to the actual subject at hand, order of operations with respect to implicit multiplication, and how the word "wrong" can be used for effect.

Me, I was actually referring to the various pronunciations I've heard (IPA /lɛd/, /liːd/, /lid/), and didn't even realize I was implying they were three different words.

(That's typical of how I make errors in English: I read and write a lot, and check for typos and misspellings, but since I don't speak English often, I don't get the same feedback on such oddities as one would with face-to-face discussion.  For example, English adjectives have a specific order: opinion – size – physical quality – shape – condition – age – color – pattern – origin – material – type – purpose.  So, while saying "That's a big soft old red British felt teddybear" is natural for native speakers, us non-native speakers can equally likely write just "That's a British felt old red soft big teddybear", which in a face to face discussion would show up in body language, and would help a non-native speaker to adjust their word order until it no longer causes such a reaction.  Most English speakers follow that rule without even realizing it.)

[IanB]

English is full of traps for the unwary. For instance, this:

It's not like I'm a native English speaker myself, you see.

For most British readers, this would be understood as: "I am a native speaker of English."

You had me confused there, for a while.

[Nominal Animal]

It's not like I'm a native English speaker myself, you see.

For most British readers, this would be understood as: "I am a native speaker of English."

You had me confused there, for a while.

Definitely not my intention; apologies!   I do fail often with such phrasings, too.  Dammit.

[SiliconWizard]

English is full of traps for the unwary. For instance, this:

It's not like I'm a native English speaker myself, you see.

For most British readers, this would be understood as: "I am a native speaker of English."

Really?

[Gyro]

English is full of traps for the unwary. For instance, this:

It's not like I'm a native English speaker myself, you see.

For most British readers, this would be understood as: "I am a native speaker of English."

You had me confused there, for a while.

Well I'm British and I understood it. If anything, it reads more like American phrasing to me. 

[Scherms]

All this guff because you were off sick the day they taught mathematics at school!
 

[Nominal Animal]

Is Your Calculator WRONG?  If Not, Get a Wrongulator; It IS!
Youtube short

[bd139]

Clearly an Aliexpress purchase.